7 N ov 1 99 5 Period Doubling , Entropy , and Renormalization

This conjecture is now established for quadratic polynomials (as a consequence of [Su] or [La]) and work is in progress toward generalization for higher degree polynomials [Hu]. The interest in such a conjecture comes from Theorems A and B below (see section 2.1) and the fact that topological entropy (conceived as an invariant of topological conjugacy [AKM]) is also one way to measure the complexity of the dynamics of a map (see section 2.1): one is trying to describe how maps with simple dynamics can be deformed to maps with complicated dynamics, or, as one says, chaotic maps. Tradition, as well as the availability in this framework of a greater set of techniques, have put some emphasis on the particular case of polynomial maps, as in Conjecture A. However, the problem of the transition to chaos is more generally interesting in the category of smooth maps, in particular smooth endomorphisms of the interval, for which we recall the following: